Geometric class field theory

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In mathematics, geometric class field theory is an extension of class field theory to higher-dimensional geometrical objects: much the same way as class field theory describes the abelianization of the Galois group of a local or global field, geometric class field theory describes the abelianized fundamental group of higher dimensional schemes in terms of data related to algebraic cycles.

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields and "global fields" such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology. Given such a field, an absolute value can be defined on it. There are two basic types of local fields: those in which the absolute value is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.

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